Dipolar Recoupling: Heteronuclear · 2019. 9. 18. · DIPOLAR RECOUPLING: HETERONUCLEAR 3 Table 1 Typical magnitudes of selected one-bond dipolar couplings in peptides and proteins

Dipolar Recoupling:HeteronuclearChristopher P. JaroniecOhio State University, Columbus, OH, USA

1 Introduction 12 MAS Hamiltonian 33 Heteronuclear Dipolar Recoupling in Spin Pairs 44 Heteronuclear Dipolar Recoupling in Multispin Systems 85 Conclusions 166 Related Articles 167 References 17

1 INTRODUCTION

Currently, high-resolution and high sensitivity NMR spectraof various types of polycrystalline solid samples containinglow-γ spin-1/2 nuclei, such as 31P, 13C, and 15N, can berecorded on a relatively routine basis. This is made possibleby the combined effects of (i) high static magnetic fieldsup to ∼22 T (950 MHz proton frequency), (ii) magic-anglespinning (MAS)1 – 3 which involves the rapid rotation of thesample (at frequencies up to ∼70 kHz) about an axis tiltedrelative to the magnetic field at an angle tan−1

√2 ≈ 54.74

(see Magic-Angle Spinning and Rotating Solids), (iii) cross-polarization (CP) from protons4,5 (see Cross Polarizationin Solids), and (iv) efficient proton decoupling6 – 8 (seeHeteronuclear Decoupling in Solids). MAS facilitates theacquisition of high-resolution solid-state nuclear magneticresonance (SSNMR) spectra by effectively averaging theanisotropic parts of nuclear spin interactions that can berepresented by second-rank tensors9: chemical shifts andthrough-space dipole–dipole couplings for spin-1/2 systems(see Internal Spin Interactions and Rotations in Solids).Consequently, in order for these interactions – which arethe primary source of information about three-dimensionalmolecular structure – to be detected and quantified underMAS conditions, they must be reintroduced or “recoupled”into the spectra. This type of recoupling can generally beachieved by the concurrent application of radiofrequency (RF)pulse sequences that appropriately manipulate the spin partsof the chemical shift and dipolar Hamiltonians and interferewith their spatial averaging due to MAS – note, however,that, in certain cases, purely MAS-driven recoupling is alsopossible.10 – 13

As thoroughly discussed elsewhere14 – 18 (see REDOR andTEDOR; Homonuclear Recoupling Schemes in MAS NMR),beginning in the late 1980s, a number of major advances havebeen made in the development of MAS NMR pulse schemesdesigned to recouple homonuclear and heteronuclear dipolarinteractions as well as chemical shift anisotropies (CSA),with most of the initial dipolar recoupling schemes gearedtoward isolated spin-1/2 pairs, such as 13C–13C, 13C–31P,or 13C–15N, incorporated at specific sites in the sample byusing various isotopic labeling approaches. Several of these

dipolar recoupling schemes including rotational echo doubleresonance (REDOR),19,20 rotational resonance (R2),11,12 anddipolar recoupling with a windowless sequence (DRAWS)21

permit highly accurate and precise measurements of dipolarcouplings as low as ∼20–30 Hz to be performed. This yieldsquantitative, site-specific internuclear distance restraints upto ∼6–8 A (note that the range of accessible distances canbe further increased to ∼10–15 A by using selective 19Flabeling22) with uncertainties of a few tenths of an angstrom,and, as a result, these methods have had and continue tohave tremendous impact on the SSNMR characterization ofmolecular structures of challenging biological systems15,22 – 25

such as protein–protein and protein–ligand complexes,26 – 28

membrane proteins,29 – 31 surface-bound peptides,32 and high-molecular-weight peptide and protein aggregates24,33 – 35 (seeRotational Resonance in Biology; Bacteriorhodopsin andRhodopsin; Structure and Dynamics of Proteins Adsorbedat Biomaterial Interfaces).

Notwithstanding the numerous successful applications ofhomonuclear and heteronuclear dipolar recoupling techniquesto biological solids containing “magnetically dilute” spin-1/2pairs, there is a compelling motivation for pursuing analogoustypes of experiments in highly or uniformly 13C,15N (U-13C,15N) labeled molecules. The primary advantage offeredby multispin systems of this type is that a multitude of site-resolved structural restraints can, in principle, be extractedby using one or few samples, as is done routinely inthe context of modern biomolecular solution-state NMR36,37

(see Biological Macromolecules: Structure Determinationin Solution; Three- and Four-Dimensional HeteronuclearMagnetic Resonance). Indeed, in recent years, by takingadvantage of the principles of multidimensional spectroscopy38

(see Multidimensional Spectroscopy: Concepts) and improvedsample-preparation methods,39 – 42 highly resolved MAS NMRspectra were reported for a number of U-13C,15N enrichedbiological macromolecules, which paved the way for thedetailed characterization of molecular structure and dynamicsin these systems and demonstrated the general feasibilityof such studies43 – 45 (see Structure Determination of SolidProteins Using MAS and Isotopic Enrichment). To illustratethe resolution and sensitivity of SSNMR spectra that canbe obtained for U-13C,15N labeled biological moleculesof varying complexity at, by today’s standards, moderatemagnetic fields (ω0/2π ∼ 500 MHz for protons) and MASrates (ωr/2π ∼ 10 kHz), in Figure 1 we show 1-D 13C and15N CP-MAS spectra of a three-residue peptide, formyl-Met-Leu-Phe (f-MLF),46,47 and 2-D 15N–13C SPECIFIC-CP48

correlation spectra of amyloid fibrils formed by residues23–144 of human prion protein (huPrP23-144).49

In instances where the recoupling of nominally large,one-bond dipolar interactions (Table 1) is of interest, theoriginal pulse schemes developed for selectively isotopelabeled samples can often be applied directly to U-13C,15Nlabeled ones with minimal or no modifications. Examples ofsuch applications include magnetization transfer in multidi-mensional chemical shift correlation experiments,50,51 char-acterization of conformational dynamics52 – 59 (see Double-Quantum NMR Spectroscopy of Dipolar Coupled SpinsUnder Fast Magic-angle Spinning; Dipolar Coupling:Molecular-Level Mobility), and measurements of relativedipole tensor orientations, which yield protein backboneand side-chain dihedral angle restraints.60 – 74 On the other

2 DIPOLAR RECOUPLING: HETERONUCLEAR

55 45 35 25 15

L F M L M F M L* L

Ca Cb Cg Cd Ce

178 176 174 172 170

180 140 100 60 20

13C Chemical shift, w1/2π (ppm)

15N Chemical shift, w1/2π (ppm)

135 125 115 105 95

(a)

G119/G126G123

G124

G127S132

S135b

S132

S135L125A116

A120

A118A133

A113

A115L130

P137 P137d

L125b

H140V121

V122

F141

I138

I139M134

M112/R136

A117

G119 G126

G124

G123

G127

G131

G114

G131

S135A133

A118

A115

A117

A113

A116/L125

V121

V122 L130

P137

F141I138

I139/H140

M134R136

A120

G114

100

110

120

130

140

100

110

120

130

140

d(15

N)

(ppm

)

178 174 170

d(13C) (ppm) d(13C) (ppm)

65 60 55 50 45

(b) (c)

M L F

L F M

M

C'

Figure 1 (a) 1-D 13C and 15N CP-MAS spectra of formyl-U-13C,15N-Met-Leu-Phe recorded at 500 MHz 1H frequency and 8.9 kHz MAS rate.(Adapted from Ref. 47. American Chemical Society, 2000) (b) 2-D 15N–13C′ and (c) 15N–13Cα spectra of U-13C,15N–huPrP23-144 amyloidfibrils recorded at 500 MHz 1H frequency and 11.1 kHz MAS rate. Note that only residues comprising the relatively rigid core region of theamyloid fibrils are detected in these spectra. (Reproduced from Ref. 49. The National Academy of Sciences of the USA, 2008.)

hand, quantitative measurements of structurally interesting,weak dipolar interactions (i.e., dipolar coupling constant<≈ 100 Hz; internuclear distance > ≈ 3 A) within tightlycoupled spin-1/2 clusters are generally less straightforwardbecause of the potential interference from multiple direct

and indirect spin–spin couplings, which contain little usefulstructural information themselves, yet are oftentimes compa-rable to or larger than the weak dipolar couplings of inter-est – in addition to the one-bond dipolar couplings listedin Table 1, these interactions include two-bond 13C–13C

DIPOLAR RECOUPLING: HETERONUCLEAR 3

Table 1 Typical magnitudes of selected one-bond dipolar couplings in peptides and proteins

Nucleus I Nucleus S I –S distance, rIS (A) I –S Dipolar coupling, bIS/2π (Hz)1H 13C 1.12 21 5001H 15N 1.04 10 82513C 13C 1.52 216513C′ 15N 1.33 130013Cα 15N 1.45 1005

and 13C–15N dipolar couplings of ∼500 Hz and ∼200 Hz,respectively, as well as one-bond 13C–13C J-couplings(∼30–60 Hz).

In this article, we focus on heteronuclear dipolar recou-pling and discuss several recent SSNMR methods, whichalleviate some of the major problems associated with thepresence of strong dipolar and J-couplings in U-13C,15Nenriched molecules and enable the accurate and precise mea-surements of multiple weak 13C–15N dipolar couplings. Thesemethods, which include frequency-selective REDOR75,76 andseveral 3-D transferred echo double resonance (TEDOR)variants,77 – 79 are based on the well-known and closelyrelated REDOR19,20 and TEDOR80 heteronuclear recouplingschemes, specifically selected for this purpose because they areparticularly robust with respect to various experimental imper-fections including finite pulse durations, resonance offsets,and RF inhomogeneity.81 – 87 Applications of these techniquesto U-13C,15N-enriched biological solids, ranging from smallpeptides and globular proteins in the microcrystalline phaseto amyloid aggregates and membrane-associated proteins, arealso highlighted.

2 MAS HAMILTONIAN

The Hamiltonian describing a system of coupled spin-1/2nuclei placed in a strong static magnetic field and subjected toMAS and time-dependent RF fields can be written as14,16

H (t) = HD(t) + HJ + HCS(t) + HRF(t) (1)

where the HD(t), HJ, HCS(t), and HRF(t) terms representdirect dipolar (through-space) spin–spin couplings, indirect(through-bond) spin–spin couplings, chemical shift interac-tions, and applied RF fields, respectively. The individualHamiltonian terms for a homonuclear spin system are given by

HD(t) =∑i<j

ωDij (t)[3IizIjz − I i · I j ] (2)

HJ =∑i<j

ωJij I i · I j (3)

HCS(t) =∑

i

ωCSi (t)Iiz (4)

HRF(t) =∑

i

|ωRFi (t)|[Iix cos φi(t) + Iiy sin φi(t)] (5)

For heteronuclear spin pairs (and homonuclear spins, forwhich the absolute chemical shift difference greatly exceedsthe magnitude of the spin–spin interaction, i.e., |ωCS

i (t) −

ωCSj (t)| |ωD

ij (t)|, |ωJij |), the dipolar and J-coupling terms

simplify to

HD(t) =∑i<j

ωDij (t)2IizIjz (6)

HJ =∑i<j

ωJij IizIjz (7)

In equations (2–7), indices i and j refer to different nuclearspins, Ix , Iy , and Iz are the spin angular momentum operators,ωJ

ij = 2πJij where Jij is the isotropic J-coupling constant inHz (anisotropic J-coupling terms are assumed to be negligible),and ωRF

i (t) = −γiBRFi (t) and are the RF-field angular nutation

frequency (determined by the nuclear gyromagnetic ratio, γi ,and RF-field amplitude, BRF

i (t)) and phase φi(t), respectively.The time-dependent coefficients ωCS

i (t) and ωDij (t) can be

conveniently expressed as Fourier series:

ωCSi (t) =

2∑m=−2

ω(m)CSi

exp(imωrt) (8)

ωDij (t) =

2∑m=−2

ω(m)Dij

exp(imωrt) (9)

with

ω(m)λ = ωλ

isoδm,0 + δλ

D

(2)0,−m(Ωλ

PR)

− ηλ

√6

[D(2)−2,−m(Ωλ

PR) + D(2)2,−m(Ωλ

PR)]

d

(2)−m,0(βRL)

(10)

where λ = CSi or Dij , ωr is the rotor frequency in angularunits, δm,0 is the Kronecker delta, and ωλ

iso, δλ, and ηλ arethe isotropic value, anisotropy, and asymmetry parameter ofinteraction λ. The Wigner rotation matrices, D

(l)

m′,m(ΩλAB),

describe the coordinate transformation between axis systemsA and B, according to a set of three Euler angles Ωλ

AB =αλ

AB, βλAB, γ λ

AB88,89:

D(l)

m′,m(ΩλAB) = exp(−im′αλ

AB)d(l)

m′,m(βλAB) exp(−imγ λ

AB)

(11)

where d(l)

m′,m(βλAB) is the reduced Wigner matrix. The coor-

dinate systems most relevant to the description of MASNMR experiments in polycrystalline solids are denoted as P


(principal axis frame), C (crystallite-fixed frame), R (rotor-fixed frame), and L (laboratory frame). Note that, for sim-plicity, equation (10) assumes that the principal axis framecoincides with the crystallite-fixed frame.

For the dipolar coupling between spins i and j we have

ωDij

iso = ηDij = 0 (12)

δDij = bij = −(µ0

4π

) γiγj

r3ij

(13)

where bij is the dipolar coupling constant in angular frequencyunits and rij is the internuclear distance. Thus, for macroscopicsample rotation exactly at the magic angle, βRL = tan−1

√2,

the Fourier coefficients in equations (8–10) are given by

ω(0)Dij

= bij

(3 cos2 βijPR − 1)

2

(3 cos2 βRL − 1)

2= 0 (14)

ω(±1)Dij

= − 1

2√

2bij sin(2β

ijPR) exp(±iγ ij

PR) (15)

ω(±2)Dij

= 1

4bij sin2 β

ijPR exp(±i2γ

ijPR) (16)

Analogous expressions can also be obtained for the chemicalshift interaction,3,14,17 where ω

CSi

iso , δCSi , and ηCSi are, in gen-eral, nonzero. Altogether, equations (8–16) demonstrate thewell-known result that MAS effectively averages anisotropicnuclear spin interactions described by second-rank tensors1 – 3

– the time-independent Fourier coefficients, ω(0)λ , vanish when

βRL is the magic angle, and the ω(±1)λ and ω

(±2)λ coefficients

oscillating at ωr and 2ωr are averaged to zero with the periodof one and one-half rotor cycle, respectively.

3 HETERONUCLEAR DIPOLAR RECOUPLING IN

SPIN PAIRS

In this section, we briefly review several methods designedto reintroduce heteronuclear dipolar interactions under MASin isolated spin pairs, with the main focus on rotaryresonance recoupling (R3)90,91 and REDOR.19,20 These initialrecoupling schemes have not only been utilized for a varietyof diverse applications,16,22 but their introduction has alsomotivated the development of a multitude of new dipolarrecoupling methodologies – indeed, as noted above, therecoupling techniques designed to measure weak 13C–15Ndipolar couplings in U-13C,15N labeled systems, which arediscussed in the following section, are all derived from theREDOR recoupling scheme. To gain basic insight into how theapplication of RF pulse sequences can interfere with MAS toreintroduce heteronuclear dipolar couplings, in the following,we consider a simple system of two coupled nuclear spins ofdifferent types, denoted by I1 and I2, in the absence of chemicalshift and J-coupling interactions.

3.1 Rotary Resonance Recoupling

R3 involves the observation of the NMR signal for spinspecies I1, in the presence of continuous-wave RF irradiationapplied to spin species I2, where the RF nutation frequencyis set to a small integer multiple of the MAS frequency(ωRF

2 = nωr).90,91 The corresponding Hamiltonian, assumingthat ωRF

2 has phase x, is given by

H (t) = ωD12(t)2I1zI2z + nωrI2x (17)

The combined effects of MAS and rotor-synchronized RFpulse sequences on nuclear spin dynamics can be convenientlyanalyzed within the framework of the average Hamiltoniantheory (AHT)92,93 (see Average Hamiltonian Theory) – thisinvolves the transformation of H (t) into an interaction framewhere the HRF(t) term vanishes, followed by calculation ofthe effective (or average) Hamiltonian describing the internalnuclear spin interactions (i.e., chemical shifts, dipolar, andJ-couplings). For the R3 experiment, the interaction framedipolar Hamiltonian is given by

˜HD(t) = einωrt I2x HD(t)e−inωrt I2x

=2∑

m=−2

ω

(m)D12

[ei(m+n)ωrt + ei(m−n)ωrt

]I1zI2z

− iω(m)D12

[ei(m+n)ωrt − ei(m−n)ωrt

]I1zI2y

(18)

and the lowest-order effective Hamiltonian, H(0)

D , is obtainedas the average of the interaction frame Hamiltonian over thepulse sequence cycle time (one rotor period, τr, in this case):

H(0)

D = 1

τr

∫ τr

0dt

˜HD(t) (19)

Since all integrals of the form∫ τr

0 dt exp[i(m ± n)ωrt] thatappear in equations (18) and (19) vanish unless a rotaryresonance condition m ± n = 0 is satisfied, the lowest-orderaverage dipolar Hamiltonian is given by

H(0)

D = (ω(−n)D12

+ ω(n)D12

)I1zI2z − i(ω(−n)D12

− ω(n)D12

)I1zI2y (20)

This result indicates that dipolar coupling terms correspond-ing to coefficients ω(±1) and ω(±2) (cf. equations (15) and(16)) are reintroduced under MAS when the RF nutation fre-

quency equals ωr and 2ωr, respectively – otherwise H(0)

D = 0and heteronuclear decoupling is predicted at this level of AHTtreatment. For example, for n = ±1 R3 we have

H(0)

D = ωD12(cos γ 12

PR2I1zI2z − sin γ 12PR2I1zI2y) (21)

where the orientation-dependent effective dipolar coupling,ωD

12, is given by

ωD12 = − 1

2√

2b12 sin(2β12

PR) (22)


With the initial density operator ρ(0) = I1x , the evolutionunder the average Hamiltonian in equation (21) yields anobservable I1-spin signal of the following form:

S(τmix) = 〈I1x〉(τmix) = 〈cos(ωD12τmix)〉 (23)

where 〈· · ·〉 indicates the powder average, and, consequently,the magnitude of the I1 –I2 dipolar coupling constant, b12, canbe extracted in a straightforward manner by inspecting theI1-spin evolution in the time or frequency domain (Figure 2).Note that the magnitude of the dipolar interaction recoupled byR3 is independent of the Euler angle γ 12

PR – this property of therecoupling sequence, commonly referred to as γ -encoding ,94

leads to particularly pronounced dipolar oscillations as seen inFigure 2(a).

The simple analysis above clearly demonstrates the emer-gence of dipolar recoupling conditions resulting from the inter-ference between MAS and applied RF fields, and also providesa reasonable approximation to the I1-spin dynamics in caseswhere the I1 –I2 dipolar coupling is the dominant interaction.However, typically, the spin dynamics during R3 are far morecomplicated owing to their strong concurrent dependence onthe chemical shift tensor parameters of the I2-spin. This stems

−1000 −800 −600 −400 −200 200 400 600 800 10000

Frequency (Hz)

0 5 10 15 20

Mixing time, tmix (ms)

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

−0.4

−0.6

(a)

(b)

⟨I1x

⟩

Figure 2 Simulated I1-spin dipolar dephasing trajectory (a) andspectrum (b) corresponding to a n = ±1 rotary resonance recouplingexperiment. The simulation was performed by the stepwise numericalpropagation of the density operator for a heteronuclear I1 –I2 (13C–15N)spin pair with a dipolar coupling constant, b12/2π = 900 Hz, in theabsence of chemical shifts and J-coupling, as implemented within theSIMPSON program.95 The MAS rate and I2-spin RF nutation frequencywere set to 25 kHz

from the fact that R3 also reintroduces I2-spin CSA terms

(described by an effective Hamiltonian analogous to H(0)

D , butwith I1z omitted and ωD

12 replaced by ωCS2 ), which, in general,

do not commute with the I1 –I2 dipolar coupling and lead tosignificant distortions of the n = ±1 and ±2 R3 dipolar trajec-tories as well as the appearance of additional, albeit weaker,higher order (|n|> 2) rotary resonances.14,16,91 Another poten-tial disadvantage associated with R3 is its sensitivity to RFinhomogeneity.90,91

While R3 itself may oftentimes not be the method ofchoice for quantitative measurements of heteronuclear dipolarcouplings, several closely related schemes have been proposedto alleviate the problems associated with the dependence ofR3 dipolar dephasing trajectories on the CSA parameters ofthe irradiated spin as well as RF inhomogeneity.63,96,97 Forexample the SPI-R3 sequence,63 which involves the applicationof a rotary resonant RF field to the I2-spin ωRF

2 = nωr thatis also continuously phase inverted during successive rotorcycles, is relatively insensitive to both I2 CSA and RFinhomogeneity. Such rotor-synchronized ±x phase alternationof ωRF

2 has no effect on the I1zI2z term in equation (21) butrepeatedly changes the sign of the I1zI2y term, meaning thatafter every two rotor periods the latter term is averaged to zero.The lowest order average dipolar Hamiltonian for n = ±1 SPI-R3 is thus given by

H(0)

D = ωD122I1zI2z (24)

with

ωD12 = − 1

2√

2b12 sin(2β12

PR) cos γ 12PR (25)

More importantly, since the effective chemical shift Hamil-

tonian for SPI-R3, H(0)

CS ∝ I2z, now commutes with H(0)

D , ithas no influence on I1-spin dynamics to lowest order – i.e.,the I1 dipolar dephasing trajectory, S(τmix) = 〈 cos(ωD

12τmix)〉,reports primarily on the magnitude of b12. We note here thatalthough the attenuation of the effects of I2-spin CSA and RFinhomogeneity by SPI-R3 is associated with the fact that thepulse sequence is no longer γ -encoded (ωD

12 in equation (25)depends on both β12

PR and γ 12PR), which leads to somewhat less

pronounced dipolar oscillations, this does not have a significantnegative impact on the accurate determination of heteronucleardipolar couplings.

3.2 Rotational Echo Double Resonance

While R3 and related pulse schemes discussed in theprevious section are all windowless, heteronuclear dipo-lar recoupling can also be achieved by applying sequencesof discrete, rotor-synchronized 180 pulses to one or bothspin species. This idea forms the basis for the REDORtechnique developed by Gullion and Schaefer.19,20 A typ-ical implementation of REDOR, with all 180 recouplingpulses – two per rotor period – applied to the nonobserved I2spin species (usually 15N in the context of 13C–15N spin pairlabeled samples) is shown in Figure 3. Note that the I1 signalduring REDOR is generally observed as a spin-echo – there-fore, for the pulse sequence in Figure 3 the entire group of


TPPMDecouplingCP

CP

0 1 2 n n+2 2n+2Rotor

15N

13C

1H

0 tr/2 3tr/2 2tr

2tr

tr

tmix/2 tmix/2

x y x y

tp tp tp tp

Figure 3 A typical implementation of the REDOR pulsesequence.19,20 The particular implementation shown is especially rele-vant to studies of U-13C,15N labeled molecules. Narrow and wide blackrectangles correspond to 90 and 180 pulses. During the REDORexperiment a 13C spin-echo is generated following 1H–13C CP. Theecho intensity is modulated during the period τmix according to themagnitude of the 13C–15N dipolar coupling, reintroduced by applyinga train of rotor-synchronized 180 pulses on the 15N channel as indi-cated in the figure. The 15N pulse phases are usually set according to thexy-4 (xyxy) or xy-8 (xyxy yxyx ) schemes,81 which offer a high degree ofcompensation for pulse imperfections. Note that REDOR experimentsgenerally involve the acquisition of two separate spin-echo trajectories:the dipolar dephased trajectory (S) using the pulse scheme shown in thefigure, and a reference trajectory (S0) that accounts for the transverserelaxation of 13C coherences not related to 13C–15N dipolar dephasing.The S0 trajectory is typically recorded in the absence of 15N 180 pulses,although implementations that involve the application of an additional15N refocusing 180 pulse85 or the time-shifting of a part of the 15N180-pulse train can also been used76,86 (cf. Figure 5). The resultingREDOR trajectories are then displayed as S/S0 or S/S0 = 1 − S/S0

15N 180 pulses following the 180 13C pulse is time-shiftedby τr/2 relative to the pulses preceding the 13C 180 to preventthe refocusing of the recoupled 13C–15N dipolar interaction.

Assuming ideal δ-function 180 pulses for the time being,the interaction frame dipolar Hamiltonian for the REDORscheme is given by

˜HD(t) = f (t)ωD

12(t)2I1zI2z (26)

where ωD12(t) is defined in equation (9), and the function f (t)

toggles between the values of ±1 during subsequent delaysbetween 180 pulses. A straightforward calculation of thelowest order average dipolar Hamiltonian using equation (19),assuming the 180 pulses are applied on the I2 channel at τr/2intervals as shown in Figure 3, yields

H(0)

D = ωD122I1zI2z (27)

with

ωD12 = −

√2

πb12 sin(2β12

PR) sin γ 12PR (28)

Note that the form of the effective dipolar Hamiltonianfor REDOR closely resembles that for n = ±1 SPI-R3 (seeequations (24) and (25)), with the main difference beingthe magnitude of the dipolar scaling factor (

√2

π≈ 0.45 for

REDOR vs. 12√

2≈ 0.35 for SPI-R3; a higher scaling factor is

generally advantageous). This indicates that, in analogy to SPI-R3, the REDOR scheme shown in Figure 3 is also relativelyinsensitive to the recoupled I2-spin CSA.

In practice, REDOR experiments are susceptible to variousexperimental imperfections including resonance offsets andRF inhomogeneity – particularly challenging are the measure-ments of weak dipolar couplings, which require the applicationof tens to hundreds of rotor-synchronized 180 pulses. Thismajor problem was addressed shortly following the introduc-tion of REDOR via the design of xy-type 180 pulse phasingschemes (e.g., xy-4: xyxy, xy-8: xyxy yxyx , etc.; see Figure 3),which offer a very high degree of compensation for such exper-imental imperfections.81,82 More recently, additional improve-ments involving the use of composite 180 pulses have alsobeen proposed.87 Altogether, these developments have resultedin REDOR being a particularly robust heteronuclear dipolarrecoupling pulse scheme, which has been successfully appliedfor the measurements of weak dipolar couplings in a numberof complex biological systems.16,22

An additional potential concern related to the use ofREDOR-type methods, which is not a factor for windowlessrecoupling schemes, is related to the effect of finite 180 pulselengths on the recoupling performance (see Accuracy Limita-tions on Internuclear Distances Measured by REDOR). Thisis a rather important issue, given that many modern SSNMRexperiments routinely take advantage of high MAS rates inthe ∼10–40 kHz regime to achieve optimal spectral resolu-tion and sensitivity. For example, for the REDOR scheme inFigure 3 implemented with a typical I2-spin RF nutation fre-quency, ωRF

2 /2π = 50 kHz, but at an MAS rate of 30 kHzinstead of the usual 5–6 kHz, ∼60% of the rotor period wouldbe occupied by RF irradiation. The effects of finite pulses onthe spin dynamics during REDOR have been analyzed usingFloquet theory83 as well as AHT and numerical simulations.84

The overall conclusion of these studies is that finite pulseeffects do not negatively impact the performance of REDORrecoupling provided that xy-4 type 180 pulse phasing schemesare employed, which is virtually always the case. We summa-rize below the AHT treatment for REDOR xy-4 and relatedsequences, which yields analytical expressions describing thespin dynamics in the presence of finite pulse effects84 – forpurpose of comparison, we also discuss a hypothetical imple-mentation of REDOR where all pulse phases are x. Finally,an experimental demonstration of the finite pulse effects forREDOR xy-4 type schemes is provided.

In general, the interaction frame Hamiltonian for theREDOR sequence in Figure 3 in the presence of finite 180pulses of arbitrary phase can be written:

˜HD(t) = ωD

12(t)[f (t)2I1zI2z + g(t)2I1zI2x + h(t)2I1zI2y]

(29)

In analogy to equation (26), the function f (t) toggles, albeitin a continuous manner, between ±1 during subsequent delaysbetween pulses regardless of the details of the pulse phase


cycling scheme. The functions g(t) and h(t) are a directconsequence of the finite pulses – they can only be nonzeroduring the pulses and their values depend on the details of thephase cycling scheme. For example, for REDOR xy-4 bothg(t) and h(t) take on nonzero values, while g(t) is always zeroif all recoupling pulses have phase x. Although the presence ofterms proportional to operators I2x and/or I2y in the effectivedipolar Hamiltonian would generally be expected to have anegative impact on the performance of REDOR recoupling –these terms would not only significantly perturb the dipolardephasing trajectories predicted for δ-function pulses but theirappearance would also indicate the concurrent recoupling ofnoncommuting I2 CSA terms – it turns out that for REDORxy-4 and extensions thereof these transverse I2-spin terms areaveraged to zero with a cycle time of two rotor periods.The resulting lowest order average dipolar Hamiltonian forREDOR xy-4 with finite pulses is

H(0)

D = ωD122I1zI2z (30)

where

ωD12 = −

√2

πκb12 sin(2β12


Note that the effective dipolar Hamiltonian for finite pulseREDOR xy-4 is nearly identical to the Hamiltonian for δ-pulseREDOR (cf. equations (27) and (28)). The only difference isthe factor κ in equation (31), which describes the finite pulseeffects and is given by

κ = cos(ϕπ/2)

1 − ϕ2(32)

where ϕ is the fraction of the rotor period occupied by thepulses:

ϕ = 2τp

τr(33)

For δ-pulse REDOR we have ϕ = 0, which gives κ = 1,and, as expected, equation (31) reduces to equation (28). At theother extreme, in the case of windowless RF irradiation (i.e.,ϕ = 1) we obtain limϕ→1 κ = π/4. This yields the effectivedipolar coupling of

ωD12 = − 1

2√

2b12 sin(2β12


which is analogous to the expression obtained for SPI-R3 (cf.equation (25)). For a typical intermediate case involving rapidMAS (e.g., ωRF

2 /2π = 50 kHz and ωr/2π = 30 kHz), we haveϕ ≈ 0.61 and κ ≈ 0.92 – i.e., AHT predicts that the dipolarcoupling is reduced by only ∼8% relative to that expectedin the δ-pulse limit. Given that this additional scaling of thedipolar coupling can be readily accounted for in simulationsby simply specifying the value of ϕ, the finite pulse effects forREDOR xy-4 and its extensions are relatively harmless.

For comparison, we briefly consider below a finite pulseversion of the REDOR scheme where all 180 pulseson the I2 channel have phase x. In this case, transverseI2-spin terms appear in the average dipolar Hamiltonian,

which leads to significant perturbations of the REDOR dipolardephasing trajectories as well as increased dependence of thesetrajectories on I2 CSA parameters. The lowest order averagedipolar Hamiltonian in this case is given by

H(0)

D = −√

2

πκb12 sin(2β12

PR)

× [sin γ 12PR2I1zI2z + ϕ cos γ 12

PR2I1zI2y] (35)

which leads to the observable signal of the form

S(τmix) = 〈cos(√

Ω2 + Φ2τmix)〉 (36)

with

Ω = −√

2

πκb12 sin(2β12


Φ = −√

2

πϕκb12 sin(2β12

PR) cos γ 12PR (38)

Note that in the case of windowless x-phase RF irradiation(ϕ = 1, κ = π/4) the observable signal in equations (36–38)reduces to the γ 12

PR -independent expression obtained for n =±1 R3 (cf. equations (22) and (23)).

In Figure 4, we provide an experimental verification ofthe finite pulse effects for REDOR xy-4 type sequences.84

Figure 4(a) shows representative experimental and sim-ulated REDOR S/S0 trajectories for 2-13C,15N-glycine,recorded under experimental conditions where 20% (ϕ = 0.2;ωRF

2 /2π = 50 kHz and ωr/2π = 10 kHz) and 61% (ϕ = 0.61;ωRF

2 /2π = 25 kHz and ωr/2π ≈ 15.1 kHz) of the rotor periodis occupied by RF pulses. These trajectories clearly demon-strate the minor scaling of the dipolar oscillation frequencypredicted above by AHT. Figure 4(b) shows the quantitativeanalysis of these data, where we plot the effective dipolar cou-pling extracted from several REDOR dephasing trajectoriesrecorded for ϕ values in the range 0.1 to 0.61. The exper-imental dipolar couplings are generally found to be in goodagreement with the corresponding values predicted by the AHTanalysis.

3.3 Symmetry-Based Pulse Sequences

Recently, starting with a scheme designed to achieve broad-band γ -encoded homonuclear 13C–13C double-quantum dipo-lar recoupling,98 Levitt and coworkers have introduced a fam-ily of general rotor-synchronized symmetry-based recouplingpulse sequences99 – 101 (see Symmetry-Based Pulse Sequencesin Magic-Angle Spinning Solid-State NMR). In the contextof heteronuclear dipolar recoupling, some of these symmetry-based sequences can be viewed as extensions of the R3 andREDOR schemes. The two most widely explored symmetryclasses are denoted CNν

n and RNνn , where N, n, and ν are

referred to as the symmetry numbers of the pulse sequence.In the case of CNν

n sequences, a total of N C-elements orcycles, each corresponding to a rotation of nuclear spins by aninteger multiple of 360, are incorporated into n rotor periods.Concurrently, the RF phases of subsequent C-elements are


1.0

0.8

0.6

0.4

0.2

0.0

S/S

0

0 1 2 3 4 5 6

Mixing time, tmix (ms)

Simulation

j = 0.20

j = 0.61

900

850

800

750

700

Effe

ctiv

e di

pola

r co

uplin

g (H

z)

0.0 0.2 0.4 0.6 0.8 1.0

j = 2tp/tr

(a)

(b)

Figure 4 Finite pulse effects in REDOR experiments. (a) Repre-sentative experimental and simulated REDOR S/S0 trajectories for2-13C,15N-glycine. The trajectories shown were recorded under exper-imental conditions, where 20% (ϕ = 0.20; yellow circles) or 61%(ϕ = 0.61; blue circles) of the rotor cycle is occupied by the REDOR180 pulses on the 15N channel. (b) Plot of the measured effective15N–13Cα dipolar couplings, beff

IS/2π, in 2-13C,15N-glycine as a func-tion of the fraction of the rotor cycle occupied by REDOR pulses,ϕ. The error bars are ±2σ . The solid line corresponds to a beff

IS

vs. ϕ curve predicted using average Hamiltonian theory, beffIS/2π =

(bIS/2π) × cos(ϕπ/2)/(1 − ϕ2), where bIS/2π = 894 Hz is the dipo-lar coupling expected in the δ-pulse limit. (Adapted from Ref. 84. Elsevier Science, 2000.)

incremented with respect to each other by the angle 2πν/N .The implementation of RNν

n sequences is similar to CNνn

sequences in that N R-elements are accommodated within n

rotor periods. However, each R-element induces a 180 ratherthan a 360 rotation of the nuclear spins, and the RF phaseis alternated between the values of ±πν/N rather than beingrepeatedly incremented.

While a detailed discussion of the various symmetry-based pulse schemes is beyond the scope of this article, thefundamental principle behind sequences of this type is that,by satisfying periodic symmetry relationships between themechanical sample rotation due to MAS and RF rotationsimposed by the pulse sequence, the space and spin trajectoriesfor the different nuclear spin interactions (represented byquantum numbers l and m, and λ and µ, respectively) canbe synchronized to generate an average Hamiltonian thatsuppresses all l, m, λ, µ combinations except for the desiredones. This means that it is possible to design pulse schemes,which permit highly selective recoupling of the nuclear spin

interactions of interest. For example, we have noted above thatin addition to recoupling heteronuclear dipolar couplings bothSPI-R3 and REDOR schemes also recouple the CSA of theirradiated spins. While such CSA recoupling is unavoidablefor single-channel pulse sequences designed to achieveheteronuclear dipolar recoupling, both REDOR and SPI-R3

can also reintroduce homonuclear dipolar couplings betweenthe irradiated spins, which may interfere with heteronucleardipolar evolution in some spin systems. Certain symmetry-based pulse sequences (R121

3, R1614, etc.; see Symmetry-Based

Pulse Sequences in Magic-Angle Spinning Solid-State NMR

for details) generate an effective dipolar Hamiltonian H(0)

D ∝I1zI2z that is analogous to that for REDOR and SPI-R3, butwith concurrent suppression of homonuclear dipolar couplings.An alternative approach to suppressing homonuclear dipolarcouplings in REDOR and SPI-R3 experiments, based on thesymmetries C31

3 and C717, has also been proposed.102

4 HETERONUCLEAR DIPOLAR RECOUPLING IN

MULTISPIN SYSTEMS

While measurements of interatomic distances in heteronu-clear 13C–15N spin pairs using SSNMR methods described inthe section “Heteronuclear Dipolar Recoupling Spin Pairs” arerelatively routine nowadays,16,22 the extension of these exper-iments to U-13C,15N labeled systems has generally not beenstraightforward owing to the presence of multiple homonu-clear and heteronuclear spin–spin couplings. This problem isparticularly exacerbated for measurements of weak 13C–15Ndipolar couplings, which correspond to structurally interestingdistances of greater than ∼3 A. For example, as discussedin more detail below, if the REDOR scheme in Figure 3were applied to a molecule containing multiple coupled 13Cand 15N nuclei, the resulting 13C dephasing trajectories wouldbe dominated by the strongest 13C–15N dipolar interactions.Furthermore, the simultaneous evolution of 13C magnetizationunder the ∼30–60 Hz one-bond 13C–13C J-couplings wouldlead to an additional modulation of these trajectories and gen-erate antiphase 13C coherences, especially for long mixing

TPPMDecouplingCP

CP

0 1 2 n n+2m 2(n+m)Rotor

15N

13C

1H

2mtr

tmix/2 tmix/2p

p

Figure 5 Frequency-selective REDOR pulse sequence.76 The recou-pling of individual 13C–15N spin pairs in U-13C,15N labeled moleculeswith concurrent suppression of 13C–13C J-couplings is achieved byusing a pair of rotor-synchronized frequency-selective (e.g., Gaussian-shaped) 180 pulses applied simultaneously on the 13C and 15N channelsas indicated in the figure. The REDOR dipolar dephasing S trajectoryis recorded using the pulse scheme as shown, and the reference S0 tra-jectory is recorded in the absence of the selective 15N pulse. See thecaption to Figure 3 for additional details of the REDOR pulse sequence


g

d2

Phe

g

z

d1

e1e2

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

0 5 10 15 20

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

0 5 10 15 20

ExperimentSimulation



Residuals (x5)

Residuals (x5)Residuals (x5)

Mixing time (ms)

Mixing time (ms)Mixing time (ms)

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

0 5 10 15 20

S/S

0

S/S

0

S/S

0

g

b

bb

a

a

d

d

da

e

3.12 Å

3.64 Å

4.12 Å

Met

Leu

Met(Cb)–Leu(N) Met(Cb)–Phe(N)

Leu(Cd)–Leu(N)

(a) (c)

(b) (d)

Figure 6 Representative frequency-selective REDOR measurements of 13C–15N distances in formyl-U-13C,15N-Met-Leu-Phe.76,111 (a) Structuralmodel of f-MLF based on the published X-ray structure of the methyl ester analog, f-MLF–OMe114 (no X-ray structure is available for f-MLF),displaying the distances determined in panels (b)–(d). Experimental FS-REDOR S/S0 curves and simulations are shown for (b) Met(Cβ)–Leu(N),(c) Leu(Cδ)–Leu(N), and (d) Met(Cβ)–Phe(N), and correspond to distances of 3.12 ± 0.03, 3.64 ± 0.09, and 4.12 ± 0.15 A, respectively. Datawere recorded at 500 MHz 1H frequency and 10.0 kHz MAS rate. For the measurements shown, resonant frequency-selective Gaussian pulses of2 ms (Met 13Cβ), 4 ms (Leu 13Cδ), and 10 ms (Leu or Phe 15N) were used to selectively recouple appropriate spin pairs. The measurements wereperformed in a sample prepared by cocrystallizing f-U-13C,15N-MLF with natural abundance f-MLF in a 1:9 ratio, to minimize the interferencefrom intermolecular 13C–15N couplings. (Reproduced from Ref. 111. The National Academy of Sciences of the USA, 2002.)

7

6

5

4

3

2

NM

R d

ista

nce,

f-M

LF (

Å)

2 3 4 5 6 7

X-ray distance, f-MLF–OMe (Å)(a) (b)

Figure 7 (a) Comparison of 13C–15N distances in formyl-U-13C,15N-Met-Leu-Phe measured by using frequency-selective REDOR76,111 with thecorresponding X-ray distances in f-MLF–OMe.114 The uncertainties in NMR distances correspond to ±2σ . (b) Representative subset of f-MLFstructures (PDB entry 1Q7O) calculated on the basis of SSNMR torsion angle and FS-REDOR 13C–15N distance restraints. (Reproduced from Ref.111. The National Academy of Sciences of the USA, 2002.)


times (τmix ∼ 10–15 ms), as well as significant lineshapeperturbations caused by the refocusing of antiphase coher-ences into observable 13C magnetization during detection.75

Although several extensions of the REDOR technique that canreport on the different 13C–15N dipolar couplings present inmultispin systems have been proposed,103 – 108 these schemesare not generally applicable to U-13C,15N peptides and pro-teins – this is primarily related to the fact that most ofthese experiments involve the application of 180 pulse trainson the 13C channel, which would also reintroduce the large∼2 kHz one-bond 13C–13C dipolar couplings that are normallyefficiently averaged by rapid MAS.109,110

In this section, we discuss several recently developedSSNMR techniques that alleviate the aforementioned problemsand facilitate the accurate and precise measurements ofmultiple weak heteronuclear 13C–15N dipolar couplings inU-13C,15N labeled molecules. These techniques are based onthe highly experimentally robust REDOR scheme discussed inthe section “Rotational Echo Double Resonance”. AlthoughREDOR does not formally suppress homonuclear dipolarcouplings between the irradiated 15N spins as noted in thesection “Symmetry-based Pulse Sequences”, this effect canbe safely neglected when considering the evolution of 13Cmagnetization for U-13C,15N enriched polypeptides, wheretypical 15N–15N dipolar couplings are small, bNN/2π <

50 Hz, and the amide 15N nuclei resonate in a relativelynarrow frequency range. Moreover, since the basic principlesbehind these new techniques are quite general, alternateimplementations employing pulse sequences that generatean analogous REDOR-like effective heteronuclear dipolar

Hamiltonian, H(0)

D ∝ I1zI2z (e.g., SPI-R3 or the appropriatesymmetry-based schemes), can also be readily envisioned.

4.1 Frequency-Selective REDOR

The effective Hamiltonian describing the spin–spin cou-plings within a system of n 13C spins and m 15N spins duringthe REDOR pulse sequence can be written:

H =n∑

i=1

m∑j=1

ωDij 2CizNjz +

∑i<j

πJij 2CizCjz (39)

where operators C and N represent 13C and 15N spins,respectively, Jij is the 13C–13C J-coupling constant andthe expression for ωD

ij , assuming that the principal axisframe for each interaction coincides with the crystallite-fixedframe, is given in equation (31). The evolution of transversemagnetization for the ith 13C spin (i.e., ρ(0) = Cix) under thiseffective Hamiltonian yields the observable powder-averagedsignal:

S(τmix) = 〈∏j

cos(ωDij τmix)〉

∏j =i

cos(πJij τmix) (40)

Since the observable 13C coherences evolve simultaneouslyas products of cosine terms involving all relevant spin–spincouplings, the strongest couplings tend to dominate the evolu-tion making the accurate quantification of weak heteronucleardipolar couplings virtually impossible.

1.0

0.8

0.6

0.4

0.2

S/S

0

10 15 20

4.0 Å

4.5 Å

5.0 Å

5.5 Å


D85

1.0

0.8

0.6

0.4

0.2

S/S

0

10 15 20

4.0 Å

4.5 Å

5.0 Å

5.5 Å


D212

Mixing time (ms)

(a)

(b)

(c)

CF B

GD85

D212

K216

Cg

Cg

Nz

Figure 8 Frequency-selective REDOR distance measurements fromthe Schiff base (Lys-216 Nζ ) to (a) Asp-85 Cγ and (b) Asp-212 Cγ

in light-adapted U-13C,15N labeled bacteriorhodopsin. Spectra wererecorded at 317 MHz 1H frequency and 6.5 kHz MAS rate; the totalexperiment duration was ∼10 days. The best-fit distances extracted fromthe experimental FS-REDOR S/S0 trajectories are 4.7 ± 0.3 A for Asp-85 and 4.9 ± 0.5 A for Asp-212 – the corresponding distances in severalX-ray structures of bR (PDB entries 1BRR,117 1QHJ,118 and 1C3W119)are in the range 4.3–5.0 A for Asp-85 and 4.0–4.4 A for Asp-212. (c)Structural model of the bR active site based on X-ray diffraction studies,showing α-helices B, C, F, and G, the chromophore (comprising retinalwith its Schiff base linkage to Lys-216), and the Asp-85 and Asp-212side chains. The distances between the Schiff base 15N and Asp 13Cγ ,measured by FS-REDOR, are indicated by dotted lines. (Adapted fromRef. 115. American Chemical Society, 2001.)

Given that the effective Hamiltonian in equation (39) isa sum of commuting bilinear terms, it can be refocusedby using spin-echo techniques. This idea forms the basisfor the frequency-selective rotational echo double resonance(FS-REDOR) scheme75,76 shown in Figure 5. FS-REDORconsists of a pair of rotor-synchronized, frequency-selective


(e.g., Gaussian-shaped) 180 pulses applied simultaneously onthe 13C and 15N channels and bracketed by two identicalREDOR periods, of length τmix/2 each, during which all13C–15N dipolar and 13C–13C J-couplings evolve. Assumingthat the selective 180 pulses are applied at frequencies ofspins Ck and Nl , the effective Hamiltonian for the entire pulsesequence that is relevant to the evolution of spin Ck is given by

H = ωDkl2CkzNlz (41)

leading to the observable Ck signal, which is equivalent to thatexpected for conventional REDOR for an isolated 13C–15Nspin pair:

S(τmix) = 〈Ckx〉(τmix) = 〈cos(ωDklτmix)〉 (42)

These results imply that within a system of many coupled13C and 15N nuclei it is, in principle, possible to isolate asingle 13C–15N dipolar coupling of interest and suppress allother 13C–15N dipolar interactions as well as 13C–13C J-couplings, provided that the chemical shifts of the relevant13C and 15N spins are such that they can be irradiated ina frequency-selective manner. In practice, bandwidths in the±2000 to ±200 Hz regime have been obtained for 1–10 msGaussian-shaped pulses.76 Note also that the selectivity of theFS-REDOR experiment can potentially be tuned further byusing longer, weaker pulses with different shapes, albeit withan accompanying loss in spectral sensitivity due to transverse13C relaxation.

The FS-REDOR technique and extensions thereof havebeen successfully applied to several U-13C,15N enrichedamino acids and peptides,76,111 – 113 including the detectionof a critical salt bridge between the side chains of residuesAsp-23 and Lys-28 in Alzheimer’s β-amyloid fibrils.112

In Figure 6, we show several representative FS-REDORmeasurements of 13C–15N dipolar couplings (corresponding todistances in the ∼3–4 A regime) for a three-residue peptide,formyl-U-13C,15N-Met-Leu-Phe (see spectra in Figure 1a).These data clearly indicate that multiple 13C–15N dipolarcouplings and 13C–13C J-couplings, that would normallysignificantly perturb the REDOR dephasing trajectories, areefficiently suppressed by FS-REDOR – notably, markedlydifferent dipolar dephasing trajectories are obtained for theMet(Cβ) site by simply changing the frequency of the

15N selective pulse from Leu(N) (Figure 6b) to Phe(N)(Figure 6d), and both trajectories are completely free ofthe effects of the large ∼200 Hz two-bond Met(Cβ)–Met(N)coupling. Altogether, FS-REDOR measurements in f-MLFhave enabled the quantitative determination of 16 13C–15Ndistances between ∼2.5 and 6 A, with typical precision(2σ ) of ∼0.1–0.3 A. The 13C–15N distances determined bySSNMR were generally found to be in good agreement withthe corresponding distances obtained using X-ray diffractionfor the methyl ester analog of f-MLF114 as illustrated inFigure 7(a) (note that no X-ray structure is available for f-MLF). Indeed, in combination with a set of backbone andside-chain torsion angle restraints, the FS-REDOR 13C–15Ndistance restraints were used to determine a high-resolutionSSNMR structure for f-MLF111 (Figure 7b).

While the applications of FS-REDOR to the structuralstudies of small U-13C,15N enriched peptides (or largersystems, segmentally labeled with U-13C,15N amino acids)displaying well-resolved 13C and 15N spectra are relativelystraightforward, analogous studies of larger U-13C,15N labeledproteins are typically considerably more challenging owingto increased spectral crowding. Nevertheless, in certain caseswhere the resonances of primary interest are sufficiently wellresolved, FS-REDOR techniques can be extended to addressinteresting structural questions in such systems.115,116 Figure 8illustrates an application of this type, where FS-REDOR wasused to determine long-range (∼4–5 A) distances between the13Cγ side-chain carbons of two aspartic acid residues anda Schiff base 15N in the active site of a 26 kDa U-13C,15Nlabeled integral membrane protein, bacteriorhodopsin (bR).115

This FS-REDOR experiment in bR was facilitated by the factthat the Schiff base 15N has a unique chemical shift of ∼165ppm in the light-adapted form of the protein (i.e., ∼30–60 ppmdownfield of backbone amide resonances), and that Cγ signalsfor the Asp-85 and Asp-212 residues of interest are relativelywell resolved from each other as well as from Cγ resonancesarising from the remaining Asp residues.

4.2 3-D TEDOR Techniques

A more general approach toward the simultaneous measure-ment of multiple heteronuclear 13C–15N dipolar couplings inU-13C,15N labeled polypeptides with arbitrary 13C and 15Nchemical shifts is based on multidimensional heteronuclear

CP

CP TPPMDecoupling Decoupling

2tr 2tr

t1

∆ ∆d

tmix

4tmix

4tmix

4tmix

4

1H

13C

15N

Figure 9 3-D z-filtered TEDOR (ZF-TEDOR) pulse sequence.78 Narrow and wide black rectangles correspond to 90 and 180 pulses. The delayδ ensures that the total delay between the REDOR dipolar mixing periods is equal to an integer number of rotor cycles, which is required for theefficient reconversion of 13C–15N antiphase coherences into observable 13C magnetization. The z-filter periods eliminate undesirable multiplequantum and antiphase spin coherences generated by 13C–13C J-evolution – in the studies discussed here, the z-filter periods were of minimumduration ( = 200 µs) and utilized a weak RF field applied concurrently on the 1H channel (ωRF ≈ ωr) to rapidly dephase transverse 13C spincoherences. However, alternate z-filter implementations can also be employed79,122


124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

60 50 40 30 20

tmix = 3.6 ms

124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

Leu

LeuLeu

Leu Leu

Leu

60 50 40 30 20

tmix = 3.6 ms

124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

60 50 40 30 20

tmix = 10.8 ms

124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

60 50 40 30 20

tmix = 10.8 ms

124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

60 50 40 30 20

tmix = 15.6 ms

124

128

132

15N

(pp

m)

60 50 40 30 2013C (ppm)

13C (ppm)

Val

60 50 40 30 20

tmix = 15.6 ms

Vg2

Vg2Vg1VbVa

Va La

VbLbLg

Ld2Aca/Ld1

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10 Representative 2-D planes from 3-D TEDOR experiments in U-13C,15N labeled acetyl-Val-Leu corresponding to dipolar mixing timesof 3.6, 10.8, and 15.6 ms. Spectra were recorded at 500 MHz 1H frequency and 10.0 kHz MAS rate, using the pulse scheme in Figure 9 implementedwithout (a)–(c) or with (d)–(f) z-filter delays. Positive and negative contours are shown in blue and red, respectively. These spectra demonstrate thatthe use of z-filters in 3-D TEDOR experiments performed on U-13C,15N labeled molecules eliminates the detrimental effects of 13C–3C J-couplingsand leads to pure absorption-mode spectra. (Adapted from Ref. 78. American Chemical Society, 2002.)


15

12

9

6

3

0

0 6 12 188

6

4

2

0

0 6 12 18

Mixing time (ms)

Cro

ss-p

eak

inte

nsity

VCg1-VNVCg1-LNSimulation

VCg2-VNVCg2-LNSimulation

7

6

5

4

3

2

11 2 3 4 5 6 7

NM

R d

ista

nce

(Å)

X-ray distance (Å)

ac-VLf-MLF

(a) (b)

Figure 11 (a) Representative 3-D ZF-TEDOR cross-peak buildup trajectories for U-13C,15N-acetyl-Val-Leu, reporting on the dipolar couplingsbetween Val 13Cγ 1 or 13Cγ 2 and Val 15N or Leu 15N. The 13C–15N distances extracted from the trajectories are as follows: Val(Cγ 1)–Val(N): 3.1 A;Val(Cγ 1)–Leu(N): 4.7 A; Val(Cγ 2)–Val(N): 4.0 A; Val(Cγ 2)–Leu(N): 3.5 A (Reproduced from Ref. 78. American Chemical Society, 2002). (b)Comparison of 13C–15N distances in U-13C,15N-acetyl-Val-Leu and formyl-U-13C,15N-Met-Leu-Phe measured by using 3-D ZF-TEDOR78 with thecorresponding X-ray distances (f-MLF X-ray distances refer to f-MLF–OMe, cf. Figures 6 and 7). Uncertainties in the NMR distances correspondto ±10% of the measured distance (see Ref. 78 for a detailed discussion). Note that U-13C,15N labeled ac-VL and f-MLF peptides used in this studywere diluted in a 1:4 and 1:9 ratio, respectively, in corresponding natural abundance peptides, to minimize the effects of intermolecular 13C–15Ncouplings

correlation (HETCOR) spectroscopy. In the simplest imple-mentation of a 3-D HETCOR scheme, a series of 2-D 15N–13Cchemical shift correlation spectra are acquired as a function ofa 15N–13C dipolar magnetization transfer (or mixing) period –note, however, that an additional chemical shift dimension canalso be readily incorporated into these schemes for increasedspectral resolution.79 Consequently, for each resolved 13Csite, Ci , cross peaks located at frequencies (ΩNj

, ΩCi) with

j = 1, 2, . . . are observed in the 2-D spectra for Ci –Nj pairsexhibiting sufficiently strong dipolar couplings – more impor-tantly, in the context of quantitative measurements, the infor-mation about Ci –Nj distances is encoded in the full cross-peakbuildup trajectories recorded as a function of the mixing time.

Several 3-D 15N–13C HETCOR schemes based on REDORmixing and 90 13C and 15N magnetization transfer pulses havebeen proposed77 – 79,120 on the basis of the original TEDORsequences developed by Schaefer and coworkers.80,121 Ofthese 3-D TEDOR experiments, the “out-and-back” typeschemes with 13C detection, that avoid the recouplingof 13C–13C dipolar interactions by minimizing the RFirradiation on the 13C channel are the most promisingfor applications to U-13C,15N labeled molecules.77 – 79 Inaddition, for measurements of weak 13C–15N couplingsrequiring relatively long mixing times, these schemes mustbe compensated for spectral artifacts caused by homonuclear13C–13C J-evolution. One such scheme, shown in Figure 9and referred to as 3-D ZF-TEDOR,78 employs two z-filter periods to suppress undesirable antiphase and multiplequantum coherences generated by the evolution of transverse13C magnetization under one-bond 13C–13C J-couplings. Theimportance of eliminating these undesirable 13C coherences

is illustrated in Figure 10 for a model dipeptide, U-13C,15N-acetyl-Val-Leu (ac-VL). Specifically, in the absence of the z-filter periods spurious cross peaks and phase-twisted lineshapesare observed in the 2-D 15N–13C spectra, particularly forlonger mixing times (Figure 10(a)–(c)), which precludes theaccurate determination of the actual cross-peak positions andintensities. On the other hand, the insertion of z-filters asindicated in Figure 9 restores pure absorption-mode correlationspectra that can be readily analyzed for all mixing times(Figure 10(d)–(f)).

The spectra in Figure 10(d)–(f) illustrate the utilityof 3-D ZF-TEDOR for simultaneously detecting multiplestructurally interesting 13C–15N dipolar couplings in U-13C,15N enriched biomolecules. For example, the relativeintensities of correlations between valine Cγ 1 and Cγ 2 andboth amide 15N sites obtained for different mixing timesimmediately provide valuable qualitative information about therotameric state of the side chain for that residue. Quantitativeestimates of the 13C–15N distances can be obtained bymonitoring the complete cross-peak buildup trajectories, withthe Ci –Nj cross-peak intensity as a function of τmix given by

Iij (τmix) = ie−iτmix

⟨sin2

(ωD

ij τmix

2

)

×∏k =j

cos2(

ωDikτmix

2

) ⟩∏l =i

cos2(πJilτmix/2) (43)

where i is the cross-peak amplitude scaling factor, andi is the relaxation rate constant for Ci spin coherences.


I107A108

T106T106b

T106b T106g

T106a

A108a

A108b

T106a Y106a

Y105bY105a

I107a I107b I107g1

I107g2

I107d

T106g

Y105

70 60 50 40 30 20 1013C Chemical shift (ppm)

15N

Che

mic

al s

hift

(ppm

)

14

12

10

8

6

4

2

0

0 2 4 6 8 10 12 14

Mixing time (ms)

3.5 Å

3.6 Å

3.2 Å

2.6 Å

T106 b T106NT106 b I107NSimulation

Cro

ss-p

eak

inte

nsity

7

8

6

5

4

3

2

1

0

0 2 4 6 8 10 12 14

Mixing time (ms)

T106 g T106NT106 g I107NSimulation

Cro

ss-p

eak

inte

nsity

Y105

T106

A108

I107

Y105

T106

A108

I107

(a)

(b)

(c)

Figure 12 Representative 3-D ZF-TEDOR 13C–15N distance measurements for transthyretin 105–115 (amino acid sequence YTIAALLSPYS)amyloid fibrils, containing a U-13C,15N labeled four-residue stretch Y105–A108. (a) Strips from a 2-D 15N–13C correlation spectrum recorded witha mixing time τmix = 6.0 ms. Resonance assignments are based on Ref. 124. (b) Experimental and simulated trajectories for the T106(Cβ)–T106(N)and T106(Cβ)–I107(N) cross peaks as a function of the dipolar mixing time, and the relevant molecular fragment displaying the measured distances.(c) Same as panel (b) but for T106(Cγ ). (Reproduced from Ref. 125. The National Academy of Sciences of the USA, 2004.)

While 3-D TEDOR-type schemes simultaneously restore thedipolar couplings between a given 13C spin and all nearby15N nuclei, the cross-peak buildup trajectories are foundto be determined primarily by the active dipolar couplingresponsible for the appearance of the cross peak at frequencies(ΩmathrmNj

, ΩCi) in the 2-D spectra and represented by the

sine squared term in equation (43). Moreover, although 3-DZF-TEDOR suppresses most of the detrimental 13C–13C J-evolution effects, the cross-peak intensities are still modulated

by 1JCC, which reduces the spectral sensitivity and, in practice,restricts the useful 15N–13C mixing times to τmix ∼ 8–14 msdepending on the value of 1JCC. Finally, we note that thecross-peak buildup trajectories exhibit a formal dependenceon the relative orientations of the active and passive 13C–15Ndipolar couplings – this is somewhat problematic since therelative dipolar tensor orientations are generally unknownand to include them as fit parameters would be impractical.Fortunately, it is possible to use instead a simple analytical


T106 A108

A109L111

L110

P113 S115

S112

Y105

Y114

I107

Figure 13 High-resolution three-dimensional structure of transthyretin105–115 in amyloid fibrils determined using MAS SSNMR tech-niques (PDB entry 1RVS).125 The structure was determined using76 experimental restraints, recorded primarily on three peptide sam-ples that were U-13C,15N labeled in overlapping stretches of fourresidues: Y105–A108, A108–L111, and L111–Y114. The experimen-tal restraints included 35 13C–15N distances in the ∼2.5–6 A regimedetermined mostly using 3-D ZF-TEDOR, and 41 backbone torsionangle restraints (on 19 φ and ψ angles) based on dipolar tensor corre-lation techniques and isotropic chemical shifts

model based on Bessel function expansions of REDORNMR signals123 that depends only on the dipolar couplingmagnitudes and is completely free of geometric parameters.According to this model, the Ci –Nj cross-peak intensity as afunction of τmix can be approximated as78

Iij (τmix) ≈ ie−iτmix1 − [J0(

√2bij τmix/2π)]2

×∏k =j

1 + [J0(√

2bikτmix/2π)]2

×∏l =i

cos2(πJilτmix/2) (44)

where J0(x) is a Bessel function of zeroth order. For spinsystems exhibiting highly resolved spectra all cross-peaktrajectories corresponding to a given 13C site can be fitsimultaneously with a set of expressions similar to equation(44), which reduces the number of independent fit parametersand results in highest quality fits78 – however, given that themost important features of the cross-peak buildup trajectoriesare primarily determined by the active dipolar coupling, insystems exhibiting more significant spectral overlap, individualtrajectories can be modeled as discussed in Ref. 79. Finally,we note that the use of this approximate analytical model,as opposed to the analytical AHT result given in equation(43), is generally expected to be the major source of error inthe measurements of 13C–15N distances in the 3–5 A regime,with typical uncertainties on the order of ∼10–15% of themeasured distance.78

Figure 11(a) shows representative experimental and sim-ulated 3-D ZF-TEDOR trajectories for ac-VL. These dataillustrate that 13C–15N dipolar couplings corresponding todistances up to ∼5 A can be readily determined using thismethod, and that the simulation model in equation (44) pro-vides a reasonable description of the experimental cross-peakbuildup trajectories. In Figure 11(b), all 13C–15N distancesmeasured in ac-VL and f-MLF by using 3-D ZF-TEDOR78

are compared with the corresponding X-ray diffraction dis-tances – overall, the two sets of distances show a remarkablyhigh degree of agreement. To demonstrate an application of3-D ZF-TEDOR to a more challenging biological system, inFigure 12, we show representative spectra, as well as the exper-imental and simulated cross-peak buildup trajectories for a

100

110

120

130

d(15

N)

(F1,

ppm

)

T49N–T49g

K13N–L12d

D40N–V39g

V21N–A20b

T49N–g

T44N–g

T11N–g

I6N–d1

T49N–A48b

A20N–b

T11N–T11g

T51N–T51g/T53N–T53g

V29N–V29g/V54N–V54g

T49N–A48b

V21N–A20b

A20N–A20b

A26N–T25g/T55N–T55g

K13N–L12d

L5N–L5d/L7N–L7d

T44N–T44g

T16N–T16g

V54N–V54g

T18N–T18g

T25N–A24b/N35N–A34b

V39N–V39g

L7N–L7d

L5N–L5d T55N–V54g

D40N–V39g

E19N–T18g

E27N–A26b

F52N–T51g/E56N–T55g

K28N–M1e*

A48N–A48b

A24N–A23b/A24b

A23N–A23b/A34N–A34b

A26N–A26b I6N–I6d1

I6N/L7N–I6g2

tCN = 14.4 ms

30 25 20 15 10

d(13C) (F2, ppm)

2.5

2.0

1.5

1.0

0.5

0.0

Cro

ss-p

eak

inte

nsity

(a.

u.)

0 5 10 15 20 25

Mixing time, tCN (ms)

4.5

4.0

3.5

3.0

2.5

2.02.0 2.5 3.0 3.5 4.0 4.5

T49N–T49g

I6N–I6d1

Simulation

NM

R d

ista

nce

(Å)

(a)

(b)

(c) X-ray distance (Å)

Figure 14 (a) 2-D slice from a 3-D SCT-TEDOR experimenton protein GB1 corresponding to the 15N–13Cmethyl chemical shiftcorrelation spectrum recorded with a dipolar mixing time 14.4 ms.Positive and negative cross peaks are drawn in blue and red,respectively. (b) Representative trajectories of cross-peak intensity asa function of the mixing time for T49N–T49γ and I6N–I6δ1 crosspeaks, corresponding to 15N–13C distances in the ∼3–4 A range. Best-fit simulations using an analytical model are also shown ( – ). (c)Comparison of selected 15N–13C distances in GB1 determined using X-ray diffraction and 3-D SCT-TEDOR. Error bars correspond to ±10% ofthe measured distance. (Reproduced from Ref. 79. American Instituteof Physics, 2008.)

segmentally U-13C,15N labeled amyloidogenic peptide corre-sponding to residues 105–115 of transthyretin.124,125 In fact,


K50 K28

M1

Q2

M1N

K28N*

K50N

K50Nz

K28Nz*

Q2N

100

110

120

130

140

F1,

15N

fold

ed (

ppm

)

F3,13C (ppm)

F2 = 15.9 ppmtCN = 15.84 ms

17 15

(a) (b)

Figure 15 (a) Small (F1,F3)-region taken from the 3-D 15N–13Cmethyl –13C correlation spectrum of protein GB1 at the M1ε frequency in F2,showing the correlations between M1ε and the neighboring 15N nuclei. Six cross peaks were observed in this region and assigned on the basis ofthe GB1 13C and 15N chemical shifts,55 combined with the analysis of published GB1 crystal structures.129 The four intramolecular correlationscorrespond to dipolar contacts between M1ε and M1N (NH3

+), Q2N, K50N, and K50Nζ , and the two remaining cross peaks (indicated by asterisks)have been assigned to intermolecular contacts with K28N and K28Nζ . (b) Structural model of GB1 in the trigonal lattice (PDB entry 1PGB),129

which qualitatively accounts for the observed cross-peak pattern, with the relevant M1ε – N distances indicated by dotted lines. The neighboringGB1 molecules in the crystal lattice are shown in ribbon representation in cyan and yellow, and residues M1, Q2, K50, and K28 are shown in stickrepresentation (the atom-types for these residues are colored as follows: C = green, O = red, N = blue, S = yellow). (Reproduced from Ref. 79. American Institute of Physics, 2008.)

these 3-D ZF-TEDOR distance measurements provided a set ofcritical restraints used to determine the high-resolution peptidestructure in amyloid fibrils (Figure 13).125

Despite the 1JCC-modulation of cross-peak intensities,which attenuates the spectral sensitivity and somewhat limitsthe range of accessible dipolar mixing times, the 3-D ZF-TEDOR scheme shown in Figure 9 and applied to U-13C,15Nlabeled samples is highly useful owing to its ability to rapidlyprovide a large number of 13C–15N distance restraints. Inaddition, several approaches have been proposed to suppressthe 13C–13C J-modulation of cross-peak intensities in 3-D TEDOR experiments, resulting in increased sensitivityand/or resolution spectra at the expense of the numberof 13C–15N dipolar couplings that can be simultaneouslydetermined in a single experiment. These approaches includethe use of (i) selective 13C refocusing pulses,78 (ii) constant-time (CT)- or semiconstant-time (SCT)-type pulse sequenceelements of duration ∼1/1JCC to refocus the J-evolution,79

and (iii) biosynthetic labeling schemes that yield proteins fullyenriched with 15N, but 13C-labeled only at specific sites fordifferent residues126,127 – this type of “magnetic dilution”abolishes most one-bond 13C–13C J-couplings and allows the3-D ZF-TEDOR scheme to be applied with no additionalmodifications.128 As an example of this type of methodology,Figures 14 and 15 show an application of 3-D and 4D SCT-TEDOR schemes to the determination of multiple distancerestraints between amide 15N sites and side-chain 13C methylgroups of alanine, isoleucine, leucine, methionine, threonine,and valine residues in the 56-amino acid B1 immunoglobulinbinding domain of protein G (GB1) in the microcrystallinephase.79 Remarkably, these experiments provide both intra-and intermolecular 13C–15N dipolar coupling restraints, whichyield information about the side-chain dihedral angles and thepacking of protein molecules within the crystal lattice.

5 CONCLUSIONS

Recent advances in MAS SSNMR instrumentation andmethodology as well as sample-preparation protocols havefacilitated the widespread studies of highly or uniformly13C,15N enriched peptides and small to medium-sized proteins,including a number of important biological systems thatpose significant challenges for traditional high-resolutiontechniques. Measurements of long-range (greater than ∼3A) internuclear distance restraints in these systems by usingdipolar recoupling techniques have the capacity to provide site-specific atomic-level insights about their three-dimensionalstructure and intermolecular interactions. In this article, wediscussed several recent REDOR-based heteronuclear dipolarrecoupling methods, which are experimentally robust andstraightforward to implement and analyze, and which enablethe accurate and precise measurements of multiple, weak13C–15N dipolar couplings in U-13C,15N enriched peptides andproteins. In addition to the specific applications highlightedhere, a multitude of analogous applications of these andrelated methods to other biological systems can also be readilyenvisaged.

6 RELATED ARTICLES

Magic-Angle Spinning; Rotating Solids; Cross Polarizationin Solids; Heteronuclear Decoupling in Solids; REDOR andTEDOR; Homonuclear Recoupling Schemes in MAS NMR;Symmetry-Based Pulse Sequences in Magic-Angle SpinningSolid-State NMR; Structure Determination of Solid ProteinsUsing MAS and Isotopic Enrichment.


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Acknowledgments

I would like to thank V. S. Bajaj, C. Filip, R. G. Griffin, J. J. Helmus,J. Herzfeld, M. Hohwy, V. Ladizhansky, J. C. Lansing, P. S. Nadaud,

B. Reif, C. M. Rienstra, and B. A. Tounge for stimulating discussionsand their contributions to the development of the NMR techniquesdiscussed here, and C. M. Rienstra for providing a copy of a manuscriptprior to publication.

Biographical Sketch

Christopher P. Jaroniec. b. 1974. B. S. Chemistry, Kent StateUniversity, 1997; Ph. D. Physical Chemistry, Massachusetts Institute ofTechnology (with R. G. Griffin), 2003; Postdoctoral Fellow, Laboratoryof Chemical Physics, NIDDK, National Institutes of Health (with A.Bax), 2003–2006. Assistant Professor of Chemistry, The Ohio StateUniversity, 2006–present. Approximately 35 publications. Researchinterests: development of new solid-state NMR methods, applications ofmultidimensional solid-state NMR techniques to the studies of structure,dynamics and interactions of biological macromolecules.

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